MinT - Best Known (210, 210+46, s)-Nets in Base 4

Best Known (210, 210+46, s)-Nets in Base 4

(210, 210+46, 1566)-Net over F₄ — Constructive and digital

Digital (210, 256, 1566)-net over F₄, using

4¹ times duplication [i] based on digital (209, 255, 1566)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 33, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (176, 222, 1539)-net over F₄, using
    - trace code for nets [i] based on digital (28, 74, 513)-net over F₆₄, using
      - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(210, 210+46, 16447)-Net over F₄ — Digital

Digital (210, 256, 16447)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁶, 16447, F₄, 46) (dual of [16447, 16191, 47]-code), using
- constructionÂ X applied to C_e(45) âŠ‚ C_e(36) [i] based on
  1. linear OA(4²³⁹, 16384, F₄, 46) (dual of [16384, 16145, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
  2. linear OA(4¹⁹⁰, 16384, F₄, 37) (dual of [16384, 16194, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
  3. linear OA(4¹⁷, 63, F₄, 8) (dual of [63, 46, 9]-code), using
    - a code from Grasslâ€™s database [i]

(210, 210+46, large)-Net in Base 4 — Upper bound on s

There is no (210, 256, large)-net in base 4, because

44 times m-reduction [i] would yield (210, 212, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]